Assistant health minister Fiona Nash is under increasing pressure to sack her chief of staff after Prime Minister Tony Abbott failed to back her against allegations of a conflict of interest during question time.

Senator Nash’s chief of staff, Alastair Furnival, is married to Tracey Cain, the owner of lobbying firm Australian Public Affairs. The firm has represented American confectionery company Mondelez, which owns brands including Kraft peanut butter and Cadbury, and the Australian Beverages Council, which counts among its members several soft drink companies.

Mr Abbott was drawn in to the growing controversy during parliamentary question time on Thursday and noticeably failed to back Senator Nash’s judgment about her decision last week to pull the government’s healthy ratings website that was unpopular with junk food companies.

The system gives packaged foods a rating from one to five stars based on their health value.

In Parliament Labor questioned whether Senator Nash and Mr Furnival breached ministerial staff standards by not declaring his interest in the lobbying firm.

“I will take the question on notice and if there is anything more to say I will say it," Mr Abbott said.

Senator Nash confirmed on Thursday that Mr Furnival ordered the website to be taken down less than a day after it was launched. She said Mr Furnival did so under her direction.

Under questioning from the opposition, Mr Abbott said: The star arrangement in question wasn’t ready to go and that is why is wasn’t proceeded with."

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The Australian Financial Review understands the pressure on Senator Nash has sparked a row with the Prime Minister’s Office on the lack of due diligence in Mr Furnival’s appointment.

All ministerial staff were vetted by Prime Minister Tony Abbott’s chief of staff, Peta Credlin, and party heavyweight Tony Nutt before they were appointed.

On Thursday, leader of the opposition in the Senate Penny Wong accused Senator Nash of misleading the Senate when she initially failed to declare Mr Furnival’s shareholding in his wife’s lobbying firm.

This week the departmental official responsible for Commonwealth food issues, Kathy Dennis, was told she would no longer oversee the federal and state committee developing the rating system and the committee would instead report to the head of her division.

Senator Nash confirmed that Ms Dennis was the departmental official who was ordered to take the website down, though denied she had any knowledge of the change in her role.

The star rating system, set up under Labor, is controversial among packaged food companies who argue it is too simplistic.

The Australian Food and Grocery Council (AFGC), which represents the bulk of the food manufacturing industry in Australia, has argued it will impose a cost on business and its algorithms could misrepresent the true health value of foods.

On Thursday AFGC chief executive Gary Dawson described the furore surrounding Senator Nash and her chief of staff as a “political overreaction".

He said his understanding was the website was mistakenly made public before it was ready and “it was reasonable for it to be taken down while further work is done".

The star rating system has not yet appeared on the front of any packaged foods, and those involved say it is unlikely to do so until at least the end of the year. If there is not sufficient voluntary take-up of the system it will become mandatory by mid-2015.